Thanks for the example. I see that then $(x_n)$ is a bounded sequences, because $\lVert x_n\rVert_{\infty}=1$. But why does then $(Ax_n)$ not have a convergent subsequence? It is $\lVert Ax_n-Ax_m\rVert_{\infty}=\sup\limits_{t\in [0,1]}\lvert (t+1)t^n-(t+1)t^m\rvert=\sup\limits_{t\in [0,1]}\lvert (t+1)(t^n-t^m)\rvert$. I do not see that this is $\geq$ some limit, so that one cannot have a Cauchysequence.
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math12Feb 7 '13 at 12:07

$x_n(t)=(1+t)t^n$ converges pointwise to the function $x(t)=0$ if $0\le t<1$, $x(1)=2$. Since $x$ is not continuous, the convergence is not uniform. The same is true of any subsequence.
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Julián AguirreFeb 7 '13 at 13:22